Question

state work energy theorem

Answer 1

          WORK ENERGY THEOREM OR WORK ENERGY PRINCIPLE 

It states that work done by the net force acting on a body is equal to the change produced in the kinetic energy of the body.

WORK ENERGY THEOREM FOR A CONSTANT FORCE.
 
When a force F acting on a body of mass'm' produces accleration 'a' in it. After covering distance s, suppose the velocity of the body changes from u to v. We use the equation of motion v² - u² = 2as

 Multiply both sides by 1/2 m , we get

           1/2 mv² - 1/2 mu² = mas

By Nwtwon's second law,

∴            1/2 mv² - 1/2mu² = Fs

                   [ Kf - Ki = W ]                                              { ∴ W = Fs }

where Kf and Ki are the final and initial kinetic energies of the body.


        WORK ENERGY THEOREM FOR A VARIABLE FORCE

Suppose a variable force F acts on a body of mass m and produces displacement 'ds' in its own direction (θ = 0°)

  Then the small work done is 

     dW = F .ds = Fds cos 0° = Fds

The time rate of change of kinetic energy is 

       dK/dt = d/dt [ 1/2mv²]

             = 1/2 × 2v × m dv/dt

              = m dv/dt (v) = (ma) v

                                                                      [ ∵ force, F = ma and v = ds/dt ]
               dK/dt = F. ds/dt

Thus,       dK = F.ds

Integrating from the initial position (xi) to final position (xf)

                            ∫ kf ki dK =  ∫xf xi Fds

where Ki and Kf are the initial and final kinetic energies corresponding to xi and xf

OR,            Change in kinetic energy, Kf - K = ∫xf xi Fds

   ∴       Kf - Ki = work done on the body (W)

                      = increase in KE of body

This proves the work done for a variable force.



Hope it helps you !